Properties

Label 4-1037575-1.1-c1e2-0-1
Degree $4$
Conductor $1037575$
Sign $1$
Analytic cond. $66.1566$
Root an. cond. $2.85195$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 7-s − 5·9-s − 3·11-s + 10·13-s + 12·16-s + 6·17-s + 4·19-s − 12·23-s + 25-s − 4·28-s + 20·36-s + 4·37-s − 24·41-s + 12·44-s + 49-s − 40·52-s + 24·53-s + 16·61-s − 5·63-s − 32·64-s − 8·67-s − 24·68-s + 4·73-s − 16·76-s − 3·77-s + 16·81-s + ⋯
L(s)  = 1  − 2·4-s + 0.377·7-s − 5/3·9-s − 0.904·11-s + 2.77·13-s + 3·16-s + 1.45·17-s + 0.917·19-s − 2.50·23-s + 1/5·25-s − 0.755·28-s + 10/3·36-s + 0.657·37-s − 3.74·41-s + 1.80·44-s + 1/7·49-s − 5.54·52-s + 3.29·53-s + 2.04·61-s − 0.629·63-s − 4·64-s − 0.977·67-s − 2.91·68-s + 0.468·73-s − 1.83·76-s − 0.341·77-s + 16/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037575 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037575 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1037575\)    =    \(5^{2} \cdot 7^{3} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(66.1566\)
Root analytic conductor: \(2.85195\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1037575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1037575,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.013510185\)
\(L(\frac12)\) \(\approx\) \(1.013510185\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 - T \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.173218716258096998513080133033, −7.980947296320422939728903060714, −7.60740517353141618382825181997, −6.59955686816645115103446073239, −6.08883603672116555036225615676, −5.60394097501507294000417794620, −5.47907564040983295563221596641, −5.18966821969300542060553275505, −4.42014878811791808583679898499, −3.65361548167228953911096455855, −3.61689003045514298236484607491, −3.25332243927866569488772238304, −2.17829312401461503900585211598, −1.21536716952659975926807713910, −0.55164240983423725757689940428, 0.55164240983423725757689940428, 1.21536716952659975926807713910, 2.17829312401461503900585211598, 3.25332243927866569488772238304, 3.61689003045514298236484607491, 3.65361548167228953911096455855, 4.42014878811791808583679898499, 5.18966821969300542060553275505, 5.47907564040983295563221596641, 5.60394097501507294000417794620, 6.08883603672116555036225615676, 6.59955686816645115103446073239, 7.60740517353141618382825181997, 7.980947296320422939728903060714, 8.173218716258096998513080133033

Graph of the $Z$-function along the critical line